Mereologies in Computing Science Uppsala: Thursday, 11 November 2010 - PowerPoint PPT Presentation

1 Mereologies in Computing Science Uppsala: Thursday, 11 November 2010 Dines Bjørner � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010

2 1. Abstract 1. Abstract In this talk we solve the following problems: • we give a formal model of a large class of mereologies, – with simple entities modelled as parts – and their relations by connectors; • we show that that class applies to a wide variety of societal infrastructure component domains; • we show that there is a class of CSP channel and process structures that correspond to the class of mereologies where – mereology parts become CSP processes and – connectors become channels; – and where simple entity attributes become process states. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

3 1. Abstract • We have yet to prove to what extent the models satisfy – the axiom systems for mereologies of, for example, (Casati&Varzi 1999) – and a calculus of individuals (Bowman&Clarke 1981). • Mereology is the study, knowledge and practice of part-hood relations: – of the relations of part to whole and – the relations of part to part within a whole. • By parts we shall here understand simple entities — of the kind illustrated in this talk. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010

4 1. Abstract • Manifest simple entities of domains – are either continuous (fluid, gaseous) – or discrete (solid, fixed), and if the latter, then ∗ either atomic ∗ or composite. – It is how the sub-entities of a composite entity ∗ are “put together” ∗ that “makes up” a mereology of that composite entity — at least such as we shall study the mereology concept. • In this talk we shall study some ways of modelling the mereology of composite entities. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

5 1. Abstract • One way of modelling mereologies is using – sorts, – observer functions and – axioms (McCarthy style), • another is using CSP . � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010

6 2. Introduction 2. Introduction 2.1. Physics and Societal Infrastructures 2.1.1. Physics • Physicists study that of nature which can be measured – within us, – around us and – between ‘within’ and ‘around’! • To make mathematical models of physics phenomena, – physics has helped develop and uses mathematics, – notably calculus and statistics. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

7 2. Introduction 2.1. Physics and Societal Infrastructures 2.1.1. Physics • Domain engineers primarily studies societal infrastructure components which can be – reasoned about, – built and – manipulated by humans. • To make domain models of infrastructure components, domain engineering makes use of – formal specification languages, – their reasoning systems: formal testing, model checking and verification, and – their tools. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010

8 2. Introduction 2.1. Physics and Societal Infrastructures 2.1.2. In Nature 2.1.2. In Nature • Physicists turns to algebra in order to handle structures in nature. – Algebra appears to be useful in a number of applications, to wit: ∗ the abstract modelling of chemical compounds. – But there seems to be many structures in nature ∗ that cannot be captured in a satisfactory way by mathematics, including algebra ∗ and when captured in discrete mathematical disciplines such as sets, graph theory and combinatorics · the “integration” of these mathematically represented — structures · with calculus (etc.) — becomes awkward; · well, I know of no successful attempts. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

9 2. Introduction 2.1. Physics and Societal Infrastructures 2.1.2. In Nature • Domain engineers turns to discrete mathematics — – as embodied in formal specification languages – and as “implementable” in programming languages — in order to handle structures in societal infrastructure components. • These languages allow – (a) the expression of arbitrarily complicated structures, – (b) the evaluation of properties over such structures, – (c) the “building & demolition” of such structures, and – (d) the reasoning over such structures. • They also allow the expression of dynamically varying structures — – something mathematics is “not so good at” ! � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010

10 2. Introduction 2.1. Physics and Societal Infrastructures 2.1.2. In Nature • But the specification languages have two problems: – (i) they do not easily, if at all, ∗ handle continuity, that is, they do not embody calculus, ∗ or, for example, statistical concepts, etc., and – (ii) they handle ∗ actual structures of societal infrastructure components ∗ and attributes of atomic and composite entities of these – – usually by identical techniques – thereby blurring what we think is an important distinction. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

11 2. Introduction 2.2. Structure of This Talk 2.2. Structure of This Talk • The rest of the talk is organised as follows. • First we give a first main, a meta-example, – of syntactic aspects of a class of mereologies. • We informally show that the assembly/unit structures indeed model structures of a variety of infrastructure components. • Then we discuss concepts of atomic and composite simple entities. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010

12 2. Introduction 2.2. Structure of This Talk • We then “perform” – the ontological trick of mapping the assembly and unit entities – and their connections – exemplified in the first main meta-example – into CSP processes and channels, respectively — – the second and last main — meta-example and now ∗ of semantic aspects of a class of mereologies. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

13 3. A Syntactic Model of a Class of Mereologies 3. A Syntactic Model of a Class of Mereologies 3.1. Systems, Assemblies, Units • We speak of systems as assemblies. • From an assembly we can immediately observe a set of parts. • Parts are either assemblies or units. • We do not further define what assemblies and units are. type S = A, A, U, P = A | U value obs Ps: (S | A) → P -set • Parts observed from an assembly are said to be immediately embedded in, that is, within , that assembly. • Two or more different parts of an assembly are said to be immediately adjacent to one another. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010

14 3. A Syntactic Model of a Class of Mereologies 3.1. Systems, Assemblies, Units Units System = Environment "outermost" Assembly C11 C32 D311 D312 B4 C12 C33 B1 C21 C31 B2 B3 A Assemblies Figure 1: Assemblies and Units “embedded” in an Environment • A system includes its environment. • And we do not worry, so far, about the semiotics of all this ! � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06

15 3. A Syntactic Model of a Class of Mereologies 3.1. Systems, Assemblies, Units • Given obs Ps we can define a function, xtr Ps , – which applies to an assembly a and – which extracts all parts embedded in a and including a . • The functions obs Ps and xtr Ps define the meaning of embeddedness. value xtr Ps: (S | A) → P -set xtr Ps(a) ≡ ′ ∈ ps } end let ps = { a } ∪ obs Ps(a) in ps ∪ union { xtr Ps(a ′ ) | a ′ :A • a • union is the distributed union operator. � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c October 30, 2010, 15:06, Uppsala Seminar, 11 Nov. 2010

16 3. A Syntactic Model of a Class of Mereologies 3.1. Systems, Assemblies, Units • Parts have unique identifiers. • All parts observable from a system are distinct. type AUI value obs AUI: P → AUI axiom ∀ a:A • let ps = obs Ps(a) in ′′ ⇒ obs AUI(p • { p ∀ p ′ ,p ′′ :P ′ ,p ′′ }⊆ ps ∧ p ′ � =p ′ ) � =obs AUI(p ′′ ) ∧ ′′ ⇒ xtr Ps(a • { a ∀ a ′ ,a ′′ :A ′ ,a ′′ }⊆ ps ∧ a ′ � =a ′ ) ∩ xtr Ps(a ′′ )= {} end � Dines Bjørner 2010, Fredsvej 11, DK–2840 Holte, Denmark c Uppsala Seminar, 11 Nov. 2010 October 30, 2010, 15:06